Question

verify rolles theorem for the function f(x)=x^3-3x^2-x+3 in [1,3]​

Answer 1

Answer:

Given f(x) = x3 – 3x2 + 2x + 5 As we know that, every polynomial function is continuous as well as differentiable. So, f(x) is continuous and differentiable on the indicated interval. Also, f(0) = 5 and f(2) = 8 – 12 + 4 + 5 = 5 i.e. f(0) = 5 = f(2) Thus, all the conditions of Rolle’s theorem are satisfied. Now, we have to show that there exist a point c in (0, 2) such that f'(c) = 0 We have f'(c) = 3c2 – 6c + 2 = 0 = 0 gives We have f'(c) = 3c2 – 6c + 2 = 0 gives ⇒ c = (6 ± √(36 – 24))/6 = (6 ± 2√3)/6 = 1 ± 1/√3 ⇒ c = 1 ±1/√3 ∈ (0, 2) Hence, Rolle’s theorem is verifiedRead more on Sarthaks.com - https://www.sarthaks.com/537326/verify-rolles-theorem-for-the-function-f-x-x-3-3x-2-2x-5-on-0-2

Step-by-step explanation:

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